Expand the given expression (x+1 x )^6 - Vidyadip

Question

Expand the given expression $\left(x+\frac{1}{x}\right)^6$

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Answer

Using binomial theorem for the expansion of $\left(x+\frac{1}{x}\right)^6$ we have
$\left(x+\frac{1}{x}\right)^6=={ }^6 C_0(x)^6+{ }^6 C_1(x)^5\left(\frac{1}{x}\right)+{ }^6 C_2(x)^4\left(\frac{1}{x}\right)^2+{ }^6 C_3(x)^3\left(\frac{1}{x}\right)^3$
$+{ }^6 C_4(x)^2\left(\frac{1}{x}\right)^4+{ }^6 C_5(x)\left(\frac{1}{x}\right)^5+{ }^6 C_6\left(\frac{1}{6}\right)^6$
$\begin{array}{l}=x^6+6 \cdot x^5 \cdot \frac{1}{x}+15 \cdot 4 x^4 \cdot \frac{1}{x^2}+20 \cdot x^3 \cdot \frac{1}{x^3}+15 \cdot x^2 \cdot \frac{1}{x^4}+6 \cdot x \cdot \frac{1}{x^5}+\frac{1}{x^6} \\ =x^6+6 x^4+15 x^2+20+\frac{15}{x^2}+\frac{6}{x^4}+\frac{1}{x^6}\end{array}$

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